Prepositions used with "Home"

I've encountered different usages of preposition + Home and I want to learn which of them are correct.

Some examples:

Back home
Back at home
Back to home

As a native speaker of American English, I would only ever say "back home" and never insert a preposition between the two. I would say, "I'm going back home" when someone asks, for example, how I will spend my vacation. In this case, "home" refers to where I was raised. If I just say "I'm going home", this could just mean the place I live now OR the place I was raised.

You can use "at" with "home" as in, "I am spending the night at home" or "Cooking at home is less expensive than eating out."

If you want to use "back to" in the above context, you might tack "my house" on the end of it.

Q: Where to you want to go after dinner?

A: Let's go back to my house.

You can also use at in this context:

Q: Where did you say the blueprints were?

A: I left them back at my house OR I left them back at the house (implies everyone knows which house is being spoken about, but you do not necessarily own it.

I cannot speak for how the above usage may differ in British English.

When "home" is behaving like an adverb expressing direction, you don't need a preposition:

I shall be arriving / going / coming / leaving home late this evening.

I'm going back home.

Note that most verbs expressing direction require a preposition (e.g. "to") before the noun, but not when the noun is "home". When the noun is "home", you don't have to use the preposition. You can use a preposition with "home" (it's not incorrect), but this usage is less common:

I'm going to home. I'm going back to home.

Once you arrive home, no more direction is suggested. In this case "at" is then the appropriate preposition to use with home:

Will you be at home tonight or are you going out? No, I'll be at home.

Reference: Prepositional use for common nouns without articles (BBC Learn it!)